Relative Equilibria for Scaling Symmetries and Central Configurations
Giovanni Rastelli, Manuele Santoprete
TL;DR
The article develops a conformal-symplectic framework for scaling symmetries in Hamiltonian mechanics, introducing conformal momentum maps, scaled cotangent lifts, and augmented potentials to characterize relative equilibria and central configurations. It unifies conformally Hamiltonian dynamics on exact symplectic manifolds with explicit cotangent-bundle constructions, and proves a Noether-type result adapted to scaling via $i_{\xi_M}\omega+(c\xi)\theta=dJ_{\xi}$. The theory specializes to cotangent bundles, providing concrete formulas for $J_{\xi}$ and the augmented Hamiltonian $H_{\xi}$, and it is applied to simple mechanical systems and the Newtonian $n$-body problem to recover classical central-configuration equations. The framework broadens the scope of central configurations and relative equilibria beyond Euclidean settings and connects scaling reductions to broader geometric mechanics tools.
Abstract
In this paper, we explore scaling symmetries within the framework of symplectic geometry. We focus on the action $Φ$ of the multiplicative group $G = \mathbb{R}^+$ on exact symplectic manifolds $(M, ω,θ)$, with $ω= -dθ$, where $ θ$ is a given primitive one-form. Extending established results in symplectic geometry and Hamiltonian dynamics, we introduce conformally symplectic maps, conformally Hamiltonian systems, conformally symplectic group actions, and the notion of conformal invariance. This framework allows us to generalize the momentum map to the conformal momentum map, which is crucial for understanding scaling symmetries. Additionally, we provide a generalized Hamiltonian Noether's theorem for these symmetries. We introduce the (conformal) augmented Hamiltonian $H_ξ$ and prove that the relative equilibria of scaling symmetries are solutions to equations involving $ H _{ ξ} $ and the primitive one-form $θ$. We derive their main properties, emphasizing the differences from relative equilibria in traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive explicit formulas for the conformal momentum map. We also provide a general definition of central configurations for Hamiltonian systems on cotangent bundles that admit scaling symmetries. Applying these results to simple mechanical systems, we introduce the augmented potential $U_ξ$ and show that the relative equilibria of scaling symmetries are solutions to an equation involving $ U _{ ξ} $ and the Lagrangian one-form $θ_L$. Finally, we apply our general theory to the Newtonian $n$-body problem, recovering the classical equations for central configurations.